STATEMENT-1 `:` A particle executing simple harmonic motion comes to rest at the extreme positions.
STATEMENT-2 `:` Displacement and velocity of SHM differ in phase by `(pi)/(2)` rad.
STATEMENT- 3 `:` Soldiers are asked to break steps while crossing the bridges.

Options

A. T.T.T.
B. F.T.F.
C. T.F.F.
D. F.F.F.

Answer

To analyze the statements provided in the question, we will evaluate each statement one by one and determine their validity.

### Step 1: Evaluate Statement 1
**Statement 1:** A particle executing simple harmonic motion comes to rest at the extreme positions.

**Explanation:**
In simple harmonic motion (SHM), a particle oscillates back and forth about an equilibrium position. At the extreme positions (maximum displacement from the mean position), the velocity of the particle is zero because it momentarily stops before reversing direction. Therefore, this statement is **true**.

### Step 2: Evaluate Statement 2
**Statement 2:** Displacement and velocity of SHM differ in phase by \(\frac{\pi}{2}\) rad.

**Explanation:**
In SHM, the displacement \(x(t)\) can be expressed as:
\[ x(t) = A \sin(\omega t) \]
where \(A\) is the amplitude and \(\omega\) is the angular frequency. The velocity \(v(t)\) is the derivative of displacement with respect to time:
\[ v(t) = \frac{dx}{dt} = A \omega \cos(\omega t) \]
This can also be expressed as:
\[ v(t) = A \omega \sin\left(\omega t + \frac{\pi}{2}\right) \]
This shows that the velocity is indeed \(\frac{\pi}{2}\) radians out of phase with the displacement. Thus, this statement is also **true**.

### Step 3: Evaluate Statement 3
**Statement 3:** Soldiers are asked to break steps while crossing the bridges.

**Explanation:**
This practice is recommended to prevent the formation of resonance in the bridge structure. When soldiers march in step, they can create oscillations that may match the natural frequency of the bridge, potentially leading to destructive resonance. By breaking step, they disrupt this synchronization. Therefore, this statement is also **true**.

### Conclusion
All three statements are true.

### Summary of Statements:
- **Statement 1:** True
- **Statement 2:** True
- **Statement 3:** True